A Langlands dual realization of coherent sheaves on the nilpotent cone

Let $G$ and $\check{G}$ be Langlands dual connected reductive groups. We establish a monoidal equivalence of $\infty$-categories between equivariant quasicoherent sheaves on the formal neighborhood of the nilpotent cone in $G$ and Steinberg-Whittaker D-modules on the loop group of $\check{G}$, as conjectured by Bezrukavnikov. More generally, we establish equivalences between various spectral and automorphic realizations of affine Hecke categories and their modules, confirming conjectures of Bezrukavnikov.